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Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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Axiomatizations of the boundary operator

The German Wikipedia page on topological boundary [1] states that the boundary operator $\partial$ can be characterized by the following four axioms: $\partial\emptyset = \emptyset$, $\partial(\partial U) \subseteq \partial…
Syd Amerikaner
  • 447
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The map $q: \mathbb{R}^3 / \{0\} \to S^2$ where $q(x) = \frac{x}{|x|}$ is a quotient map

In exercise 3.64 in Lee's topology, he claims that the fibers of $q$ are open rays in $ \mathbb{R}^3 / \{0\}$, which makes sense. He then says that it is easy to check that $q$ takes open saturated sets to open sets, which seems intuitive, but I…
Peter
  • 61
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What do we know about the intersection of co-countable and Euclidean topologies?

The Euclidean topology on $\Bbb R$ is well-understood. It is the one generated by the open intervals (or even just the open intervals with rational end-points). To some extent, we also understand the co-countable topology, which is generated by the…
Asaf Karagila
  • 393,674
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The difference between inside and outside

If I parameterise $\mathbb{R}^n$ with generalised polar coordinates $(r, \Theta)$ it is possible to partition $\mathbb{R}^n$ into three parts $$A = \{x \in \mathbb{R}^n \mid r < 1\}$$ $$B = \{x \in \mathbb{R}^n \mid r = 1\}$$ $$C = \{x \in…
Lucas
  • 1,469
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Decreasing sequence of sets

$X$ is a topological space. Let $A_n$ be a non-increasing sequence of subsets of this space: $$ A_n\supseteq A_{n+1} $$ and all $A_n$ are compact sets. Is it true that $A_\infty = \bigcap_n A_n$ is empty if and only if $A_N$ is empty for some $N$?…
SBF
  • 36,041
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Characterization of open sets in an arbitrary topological space

Here is a question from a past comprehensive exam: Let $X$ be an arbitrary topological space. Prove that $G$ is open if and only if the closure of $G \cap \overline{A}$ and the closure of $G \cap A$ are equal for all $A \subset X$.
syxiao
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If space $X$ is separable with countable dense subset $E$, then does $X\setminus E$ have to have a countable basis?

Say we have a separable space $X$, and $E$ is its countable dense subset. Let $E$ contain points $\{e_{1},e_{2},\dots e_{n}\}$. $E$ can be dense only if it contains at least one point from each base set containing any point from $X\setminus E$.…
user67803
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Is the outer boundary of a connected compact subset of $\mathbb{R}^2$ an image of $S^{1}$?

A connected compact subset $C$ of $\mathbb{R}^2$ is contained in some closed ball $B$. Denote by $E$ the unique connected component of $\mathbb{R}^2-C$ which contains $\mathbb{R}^2-B$. The outer boundary $\partial C$ of $C$ is defined to be the…
Anonymous Coward
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If $X$ is a Hausdorff space, and $x_1, x_2,\ldots, x_n$ are distinct points, do they have pairwise disjoint open neighborhoods?

A quick question about Hausdorff spaces: If $X$ is a Hausdorff space and $x_1, x_2,\ldots, x_n$ are distinct points, is it possible to find pairwise disjoint open sets $U_{x_1},\ldots, U_{x_n}$ with $x_i\in U_{x_i}$?
kiwi
  • 69
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Example of space that is separable but not a Hausdorff space.

Give an example of a topological space that is separable but not a Hausdorff space. I have not been able to discover an example, I thought of the Arens-Fort space because this is separable, since this space is countable and the same set is a…
Jhon Jairo
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Is every open subset of a manifold also a manifold?

If $M$ is a manifold, is it true that every open subset of $M$ is a manifold under the subspace topology? My definition for manifold is, for every point $x \in M$ there is a open set $U \subset M$ such that $x \in U$ and $U$ is homeomorphic to the…
user82241
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Why is this definition of topological space equivalent to the common one?

While comparing the definition of a topological space among books, I found one book [1] whose definition seems to differ from the others. Here it is: Definition: Let $X$ be a non-empty set. A class (defined by the book as a "set of sets") $T$ of…
user019828
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Shouldn’t $\tan^{-1}(\mathbb{R})$ be closed?

I’m quite confused about some topological results. I know there must be something wrong in my reasoning, but I cannot find out what is wrong here. We know that: $\mathbb{R}$ is closed (and is also opened, but that’s not what confuses me) $\tan$ is…
Jujustum
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Proving constructed homeomorphism map is continuous

I was constructing a homeomorphism from $S^2 / \langle x \sim -x \rangle$ to $\mathbb{R}P^2$, the real projective plane, defined to be the space of all lines through the origin in $\mathbb{R}^3$.A basis for $\mathbb{R}P^2$ is given by the collection…
ernesto
  • 549
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I need visual examples of topological concepts

I'm trying to understand basic concepts of topology, unfortunately I'm a very visual person, and as much documentation there is on how to come up with closed/open/clopen/etc. There are very few visual examples (using actual sets of integers, shapes,…
Lee Louviere
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